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Abstract
Institutions can provide incentives to enhance cooperation in a population where this behaviour is infrequent.This process is costly, and it is thus important to optimize the overall spending. This problem can be mathematically formulated as a multiobjective optimization problem where one wishes to minimize the cost of providing incentives while ensuring a minimum level of cooperation, sustained over time. Prior works that consider this question usually omit the stochastic effects that drive population dynamics. In this paper, we provide a rigorous analysis of this optimization problem, in a finite population and stochastic setting, studying both pairwise and multiplayer cooperation dilemmas.
We prove the regularity of the cost functions for providing incentives over time, characterize their asymptotic limits (infinite population size, weak selection and large selection) and show exactly when reward or punishment is more cost efficient. We show that these cost functions exhibit a phase transition phenomena when the intensity of selection varies. By determining the critical threshold of this phase transition, we provide exact calculations for the optimal cost of incentive, for any given intensity of selection. Numerical simulations are also provided to demonstrate analytical observations. Overall, our analysis provides for the first time a selectiondependent calculation of the optimal cost of institutional incentives (for both reward and punishment) that guarantees a minimum level of cooperation over time. It is of crucial importance for realworld applications of institutional incentives since intensity of selection is often found to be nonextreme and specific for a given population.
We prove the regularity of the cost functions for providing incentives over time, characterize their asymptotic limits (infinite population size, weak selection and large selection) and show exactly when reward or punishment is more cost efficient. We show that these cost functions exhibit a phase transition phenomena when the intensity of selection varies. By determining the critical threshold of this phase transition, we provide exact calculations for the optimal cost of incentive, for any given intensity of selection. Numerical simulations are also provided to demonstrate analytical observations. Overall, our analysis provides for the first time a selectiondependent calculation of the optimal cost of institutional incentives (for both reward and punishment) that guarantees a minimum level of cooperation over time. It is of crucial importance for realworld applications of institutional incentives since intensity of selection is often found to be nonextreme and specific for a given population.
Original language  English 

Journal  Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 
Publication status  Accepted/In press  16 Sep 2021 
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Leverhulme Research Fellowship: "Incentives for Commitment Compliance"
12/12/20 → 5/12/22
Project: Research